In actual cases, the magnetic interactions with the traveling spins may induce the change of the direction of \(\vec{M}\) .
This will pose an interesting problem of the movement of the domain wall caused by the current
as studied by Berger[1]} and Tatara et al[2]}.
We assume, in the present work, that the localized moments are heavy enough,
so that their direction is unaffected by the back-reaction from the traveling spin. The localized magnetic moments
are considered to be a source of static potential to the traveling spins.
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